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## Homework Statement

Consider a spherical cap, for which the surface area and volume is

[tex]A(a,h) = \pi(a^2 +h^2)[/tex]

[tex]V(a,h) = \frac{\pi h}{6}(3a^2 +h^2)[/tex]

What would the aspect ratio [itex]dA/dV[/itex] be?

## The Attempt at a Solution

Clearly we would have

[tex]dA = 2\pi a da + 2\pi h dh[/tex]

[tex]dV = \pi ha da + \frac{\pi}{2}(a^2+h^2) dh [/tex]

but what would the chain rule look like? Surely, it should have more terms than just

[tex]\frac{dA}{dV } = \frac{\partial A}{\partial a} \frac{\partial a}{\partial V} + \frac{\partial A}{\partial h}\frac{\partial h}{\partial V}[/tex]

, right?